Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14717 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,8)(4,17,20,9)(5,22)(6,25)(7,11)(10,23)(12,27)(13,36,18,28)(14,39)(15,30)(16,43)(19,49,50,54)(21,33,52,46)(24,35)(26,34)(29,41)(31,42)(32,51,45,53)(37,61,47,64)(38,56,48,60)(40,58)(44,57)(55,62,59,63);(1,3)(2,8)(4,18,20,13)(5,14)(6,16)(7,15)(9,28,17,36)(10,29)(11,30)(12,31)(19,47,50,37)(21,48,52,38)(22,39)(23,41)(24,40)(25,43)(26,44)(27,42)(32,55,45,59)(33,56,46,60)(34,57)(35,58)(49,61,54,64)(51,63,53,62);(1,4)(2,9)(3,13)(5,19)(6,21)(7,20)(8,28)(10,32)(11,17)(12,33)(14,37)(15,18)(16,38)(22,54)(23,51)(24,50)(25,46)(26,53)(27,52)(29,55)(30,36)(31,56)(34,45)(35,49)(39,64)(40,47)(41,62)(42,48)(43,60)(44,63)(57,59)(58,61);(1,5,7,24)(2,10,11,34)(3,14,15,40)(4,19,20,50)(6,26,27,23)(8,29,30,57)(9,32,17,45)(12,35,25,22)(13,37,18,47)(16,44,42,41)(21,53,52,51)(28,55,36,59)(31,58,43,39)(33,49,46,54)(38,63,48,62)(56,61,60,64)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301681);